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Properties

Label 1-187-187.97-r1-0-0
Degree $1$
Conductor $187$
Sign $0.877 - 0.478i$
Analytic cond. $20.0959$
Root an. cond. $20.0959$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 − 0.156i)2-s + (−0.760 − 0.649i)3-s + (0.951 − 0.309i)4-s + (−0.972 + 0.233i)5-s + (−0.852 − 0.522i)6-s + (0.649 + 0.760i)7-s + (0.891 − 0.453i)8-s + (0.156 + 0.987i)9-s + (−0.923 + 0.382i)10-s + (−0.923 − 0.382i)12-s + (0.587 − 0.809i)13-s + (0.760 + 0.649i)14-s + (0.891 + 0.453i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)18-s + (0.453 + 0.891i)19-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (−0.760 − 0.649i)3-s + (0.951 − 0.309i)4-s + (−0.972 + 0.233i)5-s + (−0.852 − 0.522i)6-s + (0.649 + 0.760i)7-s + (0.891 − 0.453i)8-s + (0.156 + 0.987i)9-s + (−0.923 + 0.382i)10-s + (−0.923 − 0.382i)12-s + (0.587 − 0.809i)13-s + (0.760 + 0.649i)14-s + (0.891 + 0.453i)15-s + (0.809 − 0.587i)16-s + (0.309 + 0.951i)18-s + (0.453 + 0.891i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(187\)    =    \(11 \cdot 17\)
Sign: $0.877 - 0.478i$
Analytic conductor: \(20.0959\)
Root analytic conductor: \(20.0959\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{187} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 187,\ (1:\ ),\ 0.877 - 0.478i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.590070870 - 0.6601693763i\)
\(L(\frac12)\) \(\approx\) \(2.590070870 - 0.6601693763i\)
\(L(1)\) \(\approx\) \(1.563556272 - 0.3062350861i\)
\(L(1)\) \(\approx\) \(1.563556272 - 0.3062350861i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.987 - 0.156i)T \)
3 \( 1 + (-0.760 - 0.649i)T \)
5 \( 1 + (-0.972 + 0.233i)T \)
7 \( 1 + (0.649 + 0.760i)T \)
13 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.453 + 0.891i)T \)
23 \( 1 + (0.382 + 0.923i)T \)
29 \( 1 + (0.0784 - 0.996i)T \)
31 \( 1 + (0.852 - 0.522i)T \)
37 \( 1 + (0.996 + 0.0784i)T \)
41 \( 1 + (-0.0784 - 0.996i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.951 + 0.309i)T \)
53 \( 1 + (0.987 - 0.156i)T \)
59 \( 1 + (-0.453 + 0.891i)T \)
61 \( 1 + (0.522 - 0.852i)T \)
67 \( 1 - T \)
71 \( 1 + (0.233 + 0.972i)T \)
73 \( 1 + (-0.0784 + 0.996i)T \)
79 \( 1 + (-0.233 + 0.972i)T \)
83 \( 1 + (-0.156 + 0.987i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.522 + 0.852i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.862548296119484144278868966757, −26.328872263447750310613148042445, −24.69047094498647150615448908216, −23.592084706626608072432363680408, −23.48323406060927709705853418954, −22.36465880822630293853694556351, −21.36294998963644156526921222256, −20.56010813574948717190892971238, −19.739005380322978188678172401976, −18.111700906836505329071008840651, −16.76809178917861244427670117555, −16.29442981812871129665336943641, −15.28316872552872886707073914301, −14.41579341972090394093864856682, −13.16812558285267931891029564981, −11.93158257685228097304263548809, −11.30808031199515186265422482744, −10.52237290123684341283405322758, −8.71791372367756947881280168243, −7.35258426159615131615707188487, −6.425792243782492116030762639155, −4.84440400811648983405982898440, −4.42933748013660185809884469327, −3.28905703726864215669027606564, −1.04092729252150686671350924192, 1.059942005577819368239081367132, 2.54924777329198491257262204889, 3.97616222185031956182086701479, 5.2660039208098320689187421416, 6.05196699872560025255550702740, 7.39203572035166446480869486778, 8.16379492199738672959083709050, 10.37152245767360716171297638021, 11.448942467682851959578583185127, 11.864840336030665092646111717718, 12.84780452495565551822720427511, 13.93987553590272306071122804935, 15.21844774385429920460647127904, 15.78431716834288453969611188780, 17.067324155474888750030304448468, 18.38452923536460435158234321578, 19.09109181378244255475270692688, 20.230483422462995245648427451113, 21.300567116234222942546914942704, 22.41954667075341523666898823046, 23.00708682274075553103135267816, 23.817696541098147485411737533072, 24.662098827356451859542531144937, 25.40705917882893466889771485901, 27.198851288954479929424511845544

Graph of the $Z$-function along the critical line